Journal of Advanced Research (2013) 4, 201–204

Cairo University

Journal of Advanced Research

SHORT COMMUNICATION

On the oscillation of higher order dynamic equations

Said R. Grace

*

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt

Received 4 March 2012; revised 2 April 2012; accepted 23 April 2012

Available online 2 July 2012

KEYWORDS

Abstract

Oscillation;

Higher Order;

Dynamic equations

aðtÞðx

We present some new criteria for the oscillation of even order dynamic equation

DnÀ1

ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rdcontinuous functions deﬁned on T.

ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

(ii) a, q: T ﬁ R+ = (0, 1) is a real-valued rd-continuous

functions, aD(t) P 0 for t 2 [t0, 1)T and

Introduction

This paper is concerned with the oscillatory behavior of all

solutions of the even order dynamic equation

nÀ1

aðtÞðxD ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

ð1:1Þ

on an arbitrary time-scale T ˝ R with Sup T = 1 and n P 2 is

an even integer.

We shall assume that:

(i) a P 1 is the ratio of positive odd integers,

* Tel.: +20 2 35876998.

E-mail address: saidgrace@yahoo.com.

Peer review under responsibility of Cairo University.

Production and hosting by Elsevier

Z

1

aÀ1=a ðsÞDs ¼ 1:

ð1:2Þ

We recall that a solution x of Eq. (1.1) is said to be nonoscillatory if there exists a t0 2 T.

Such that x(t)x(r(t)) > 0 for all t 2 [t0, 1)T; otherwise, it is

said to be oscillatory. Eq. (1.1) is said to be oscillatory if all its

solutions are oscillatory.

The study of dynamic equations on time-scales which goes

back to its founder Hilger [1] as an area of mathematics that

has received a lot of attention. It has been created in order

to unify the study of differential and difference equations.

Recently, there has been an increasing interest in studying

the oscillatory behavior of ﬁrst and second order dynamic

equations on time-scales, see [2–7]. With respect to dynamic

equations on time scales it is fairly new topic and for general

basic ideas and background, we refer to [8,9].

It appears that very little is known regarding the oscillation

of higher order dynamic equations [10–15] and our purpose

here to establish some new criteria for the oscillation criteria

for such equations. The obtained results are new even for the

special cases when T = R and T = Z.

2090-1232 ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.jare.2012.04.003

202

S.R. Grace

Theorem 2.2. Let conditions (i), (ii) and (1.2) hold and

Main results

Z

We shall employ the following well-known lemma.

Z

rðsÞ ðaðsÞÞÀ1

1

t0

Lemma 2.1. [6, Corollary 1]. Assume that n 2 N, s, t 2 T and

f 2 Crd(T, R). Then

Z

t

s

Z

t

ÁÁÁ

gn þ1

¼ ðÀ1Þn

Z

Z

Z

t!1

t

ð2:1Þ

Â

Ã

gðsÞqðsÞ À aðsÞgD ðsÞhÀa

nÀ1 ðs; t0 Þ Ds ¼ 1;

ð2:4Þ

t1

t

hn ðs; rðgÞÞfðgÞDg:

ð2:3Þ

If there exists a positive non-decreasing delta-differentiable

function g such that for every t1 2 [t0, 1)T.

lim sup

gn

1=a

qðuÞDu

Ds ¼ 1:

s

t

fðg1 ÞDg1 Dg2 Á Á Á Dgnþ1

1

then Eq. (1.1) is oscillatory.

s

We shall employ the Kiguradze’s following well-known

lemma.

Theorem 2.1. (Kiguarde’s Lemma [8, Theorem 5]),Let

n 2 N; f 2 Cnrd ðT; RÞ and sup T = 1. Suppose that f is either

n

positive or negative and fD is not identically zero and is either

nonnegative or nonpositive on [t0, 1)T for some t0 2 T. Then,

there

exist

t1 2 [t0, 1)Tm 2 [0, n)Z

such

that

n

ðÀ1ÞnÀm fðtÞfD ðtÞ P 0 holds for all t 2 [t0, 1)T with

j

(I) f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all

j 2 [0, m)z,

j

(II) ðÀ1Þmþj f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all

j 2 [m, n)z

Proof. Let x(t) be a nonoscillatory solution of Eq. (1.1) on

[t0, 1)T. It sufﬁces to discuss the case x is eventually positive

(as – x also solves (1.1) if x does), say x(t) > 0 for t P t1 P t0.

Now, we see that

D

nÀ1

aðtÞðxD ðtÞÞa 6 0 for t P t1 :

nÀ1

It is easy to see that xD ðtÞ > 0 for t P t1 for otherwise,

and by using condition (1.2) we obtain a contradiction to the

fact that x(t) > 0 for t P t1. Now, aD(t) P 0 for t 2 [t0, 1)T.

We have

D

nÀ1

D

nÀ1 a

nÀ1

aðtÞðxD ðtÞÞa ¼ aD ðtÞ xD ðtÞ þ ar ðtÞ ðxD ðtÞÞa

6 0:

The following result will be used to prove the next corollary.

Lemma 2.2. [12, Lemma 2.8].Let supT = 1 and

f 2 Cnrd ðT; RÞðn P 2Þ. Moreover, suppose that Kiguradze’s theon

rem holds with m 2 [1, n)N and fD 6 0 on T. Then there exists a

sufﬁciently large t1 2 T such that

m

fD ðtÞ P hmÀ1 ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :

This implies

nÀ1

ððxD ðtÞÞa ÞD 6 0 for t P t1 :

nÀ1

Next, we let y ¼ xD

rem 1.90] we see that

ð2:2Þ

Z

on [t1, 1)T. From Ref. [9, Theo-

1

ðy þ hlyD ÞaÀ1 dh P ayD

Z

1

The proof of the following corollary follows by an integration

of (2.2).

0 P ðya ÞD ¼ ayD

Corollary 2.1. Assume that the conditions of Lemma 2.2 hold.

Then

Thus we have yD ¼ xD 6 0 on [t1, 1)T and from Theorem 2.1, there exists an integer m 2 {1, 3, . . . , n À 1}. Such that

(I) and (II) hold on [t1, 1)T Clearly xD(t) > 0 for t P t1 and

hence, there exists a constant c > 0 such that

m

fðtÞ P hm ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :

Next, we need the following lemma see [16].

Lemma 2.3. If X and Y are nonnegative and k > 1, then

0

yaÀ1 dh

0

¼ ayaÀ1 yD :

n

xðtÞ P c

ð2:5Þ

for t P t1 :

First, we claim that m = n À 1. To this end, we assume that

Xk À kXYkÀ1 þ ðk À 1ÞYk P 0;

xD ðtÞ < 0

where equality holds if and only if X = Y,

Integrating Eq. (1.1) from t P t1 to u P t, letting u ﬁ 1 we

have

It will be convenient to employ the Taylor monomials (see

[9, Section 1.6]) fhn ðt; sÞ1

n¼0 g which are deﬁned recursively by:

Z t

hn ðs; sÞDs; t; s 2 T and n P 1;

hnþ1 ðt; sÞ ¼

nÀ2

nÀ3

and xD ðtÞ > 0

Z

nÀ1

xD ðtÞ P c ðaðtÞÞÀ1

for t P t1 :

1=a

1

for t P t1 :

qðsÞDs

t

s

where it follows that h1(t, s) = t À s but simple formulas in

general do not hold for n P 2.

Now we present the following oscillation results for Eq.

(1.1).

Integrating (2.6) from to and letting ﬁ 1 we get

nÀ2

0 < ÀxD ðtÞ 6 À

Z

t

1

Z

ðaðuÞÞÀ1

u

1=a

1

qðsÞDs

Du:

ð2:6Þ

On the oscillation of higher order dynamic equations

203

Integrating this inequality from t and using condition (2.3)

after Lemma 2.1 we arrive at the desired contradiction. It

follows from Lemma 2.2 with m = n À 1 that

DnÀ1

D

x ðtÞ P hnÀ2 ðt; t1 Þx

ðtÞ for t P t1 ;

ð2:7Þ

and by applying Corollary 2.1 with m = n À 1 instead of Lemma 2.2, we get

nÀ1

xðtÞ P hnÀ1 ðt; t1 ÞxD ðtÞ

D

D

g

g

x

wD 6 Àgq þ r wr À a r

wr :

g

g

x

Using Lemma 2.2 with m = n À 1 in (2.15), we ﬁnd

!

D

nÀ1

g

g

xD

r

wr

hnÀ2

w 6 Àgq þ r w À a r

g

g

x

D

nÀ1

DnÀ1 a

aðx Þ

xa

on ½t1 ; 1ÞT :

ð2:9Þ

g D

g

nÀ1

axD

x

ð2:10Þ

Now set

ð2:11Þ

a

X ¼ ðaðaÀ1=a ÞghnÀ2 Þaþ1

wD ðtÞ 6 ÀgðtÞqðtÞ þ aðtÞgD ðtÞðhnÀ1 ðt; t1 ÞÞÀa ;

¼

t P t1 :

Integrating this inequality from t2 > t1 to t P t2, we have

t

½gðsÞqðsÞ À aðsÞgD ðsÞhÀa

nÀ1 ðs; t1 ÞDs:

ð2:12Þ

t2

Taking upper limit of both sides of the inequality (2.12) as

t ﬁ 1 and using (2.4) we obtain a contradiction to the fact

that w(t) > 0 on [t1, 1)T. This completes the proof. h

Next, we establish the following result.

t!1

Â hnÀ2 ðs; t0 ÞgðsÞ

ð2:13Þ

x þ lhxD

ÃaÀ1

a

ða þ 1Þaþ1

1

ghnÀ2

a

ðgD Þaþ1

on ½t2 ; 1ÞT :

ð2:18Þ

Integrating (2.18) from t2 to t, we get

Z t"

gðsÞqðsÞ À

aðsÞ

1

a

ða þ 1Þaþ1 gðsÞhnÀ2 ðs;t1 Þ

#

ðgD ðsÞÞaþ1 Ds:

Taking upper limit of both sides of (2.19) as t ﬁ 1 and

using (2.13), we obtain a contradiction to the fact that

w(t) > 0 for t P t1 This completes the proof. h

Finally, we present the following interesting result.

Proceeding as in the proof of Theorem 2.1, we obtain

m = n À 1 and (2.7) and (2.8). Deﬁne w as in (2.9) and obtain

(2.10). Now from Ref. [3, Theorem 1.90],

ðxa ÞD ¼ ðaxD Þ

;

ð2:19Þ

Proof. Let x be a nonoscillatory solution of Eq. (1.1), say

x(t) > 0 for t P t1 P t0.

1Â

a !a

!aþ1

and therefore, we ﬁnd

t2

then Eq. (1.1) is oscillatory.

Z

a1=a

aghnÀ2

P 0;

aðsÞðgD ðsÞÞaþ1

aþ1

Ds ¼ 1:

and Y

in Lemma 2.3 with k ¼ aþ1

> 1 to conclude that

a

!

D

g

1

g

a

ðgD Þaþ1

r 1þ1=a

ðw Þ

À r wr þ

a

aþ1

1þ1=a

hnÀ2

g

ga hanÀ2

ða þ 1Þ

ðgr Þ

wðtÞ 6 wðt2 Þ À

ða þ 1Þ

!Àa #

t1

r

w

g

a

a

ðgD Þa

aþ1

wD 6 Àgq þ

Theorem 2.3. Let conditions (i), (ii) (1.2) and (2.3) hold. If

there exists a a positive non-decreasing delta-differentiable

function g such that for every t1 2 [t0, 1)T

1

ð2:17Þ

for t P t2 P t1 ;

Using (2.8) in (2.11), we get

Z t"

lim sup

gðsÞqðsÞ À

on ½t2 ; 1ÞT ;

!

D

g

ðaÞÀ1=a g

r

ðhnÀ2 Þðwr Þ1þ1=a

w 6 Àgq þ r w À a

g

ðgr Þ1þ1=a

ðaðxD Þa Þr þ a ðaðxD Þa ÞD

" x

#

D a

a D

DnÀ1 a r g x À gðx Þ

;

¼ Àgq þ ðaðx Þ Þ

xa ðxr Þa

!a

DnÀ1

D x

6 Àgq þ ag

:

x

Z

1=a

r 1=a

w

w

P

g

g

D

nÀ1

xa

wðtÞ 6 wðt2 Þ À

¼

and thus,

Then on [t1, 1)T, we have

wD ¼

ð2:16Þ

where hnÀ2 = hnÀ2(t, t1). Now we see that

Now, we let

w :¼ g

for t

P t2 P t1 ;

ð2:8Þ

for t P t1 :

ð2:15Þ

dh P axD

0

Z

Theorem 2.4. Let conditions (i), (ii), (1.2) and (2.3) hold. If

there exists a a positive, delta-differentiable function g such that

for every t1 2 [t0, 1)T

Z t"

lim sup

t!1

1

xaÀ1 dh ¼ axaÀ1 xD :

t1

#

1

ðaðsÞÞr ðgD ðsÞÞ2

gðsÞqðsÞ À

Ds

4a gðsÞðhnÀ1 ðs; t1 ÞÞaÀ1 hnÀ2 ðs; t1 Þ

¼ 1;

0

ð2:20Þ

ð2:14Þ

Using (2.14) in (2.10), we have

then Eq. (1.1) is oscillatory.

204

S.R. Grace

Proof. Let x be a nonoscillatoy solution of Eq. (1.1), say

x(t) > 0 for t P t1 P t0

Proceeding as in the proof of Theorem 2.3, we obtain (2.17)

which cam be rewritten as

!

D

g

ðaÞÀ1=a g

r

w 6 Àgq þ r w À a

ðhnÀ2 Þ

g

ðgr Þ1þ1=a

2. We may also employ other types of the time-scales [8,9] e.g.,

T = hZ with h > 0; qN o ; q > 1, T ¼ N 20 , etc. The detail are

left to the reader.

References

D

r 1=aÀ1

Â ðw Þ

r 2

ðw Þ

on ½t2 ; 1ÞT :

ð2:21Þ

Now, using Corollary 2.1 with m = n À 1 we have

x

nÀ1

xD

P hnÀ1 ;

implies on [t2, 1)T that (2.22)

!1Àa

DnÀ1

1=aÀ1

1=aÀ1 1=aÀ1 x

w

¼a

g

x

aÀ1

x

¼ a1=aÀ1 g1=aÀ1 DnÀ1

P a1=aÀ1 g1=aÀ1 haÀ1

nÀ1 :

x

ð2:22Þ

Using (2.22) in (2.21) we have on [t2, 1)T that

wD 6 Àgq þ

D

g

gr

À

¼ Àgq À

þ

aÀ1 r

gðh Þ h

wr À a ðaÞnÀ1r ðgr ÞnÀ2

ðwr Þ2 :

2

r

ahÀ1

nÀ2 ðhnÀ1 Þ

aÀ1

Á1=2

g

À

ððaÞr Þ1=2 gr

r

ðaÞ

ðgD Þ2

4aðhrnÀ1 Þ

6 Àgq þ

aÀ1

hnÀ2

À1Á

4a

!2

ððaÞr Þ1=2 gD

aÀ1

2ðahnÀ2 ðhrnÀ1 Þ

gÞ1=2

:

ðaÞr ðgD Þ2

gðhrnÀ1 Þ

aÀ2

hnÀ2

:

ð2:23Þ

Integrating this inequality from t2 to t, taking upper limit of

the resulting inequality as t ﬁ 1, and applying condition (2.20)

we obtain a contradiction to the fact that w(t) > 0 for t P t1.

This completes the proof.

h

Remarks

1. The results of this paper are presented in a form which is

essentially new and of high degree of generality. Also, we

can easily formulate the above conditions which are new sufﬁcient for the oscillation of Eq. (1.1) on different time-scales

e.g., T = R and T = Z. The details are left to the reader.

[1] Hilger S. Analysis on measure chain-a uniﬁed approach to

contiguous and discrete calculus. Results Math 1990;18:18–56.

[2] Braverman E, Karpuz B. Nonosillation of ﬁrst-order dynamic

equations with several delays. Adv Difference Eq 2010:22. Art.

ID 873459.

[3] Grace SR, Bohner M, Agarwal RP. On the oscillation of second

order half linear dynamic equations. J Differ Eqs Appl

2009;15:451–60.

[4] Grace SR, Agarwal RP, Bohner M, O’Regan D. Oscillation of

second order strongly superlinear and strongly sublinear

dynamic equations. Commun Nonlinear Sci Numer Simulat

2009;14:3463–71.

[5] Sahiner Y. Oscillation of second order delay differential

equations on time-scales. Nonlinear Anal 2005;63(5–7):1073–80.

[6] Bohner M. Some oscillation criteria for ﬁrst order delay

dynamic equations. Far East J Appl Math 2005;18(3):289–304.

[7] Braverman E, Karpuz B. Nonoscillation of second order

dynamic equations with several delays. Abstr Appl Anal

2011:34. Art. ID 591254.

[8] Agarwal RP, Bohner M. Basic calculus on time scales and some

of its applications. Results Math 1999;35(1–2):3–22.

[9] Bohner M, Peterson A. Dynamic Equations on Time-Scales : An

Introduction with Applications. Boston: Birkhauser; 2001.

[10] Chen DX. Oscillation and asymptotic behavior for nth order

nonlinear neutral delay dynamic equations on time scales. Acta

Appl Math 2010;109(3):703–19.

[11] Erbe L, Baoguo J, Peterson A. Oscillation of nth order

superlinear dynamic equations on time scales. Rocky

Mountain J Math 2011;41(2):471–91.

[12] Erbe L, Karpuz B, Peterson A. Kamenev-type oscillation

criteria for higher order neuyral delay dynamic equations. Int

J Differ Eq, in press (2012.-IJDE-1106).

[13] Karpuz B. Asymptotic behavior of bounded solutions of a class

of higher-order neutral dynamic equations. Appl Math Comput

2009;215(6):2174–2183,.

[14] Karpuz B. Unbounded oscillation of higher-order nonlinear

delay dynamic equations of neutral type with oscillating

coefﬁcients. Electon J Qual Theo Differ Eq 2009;9(34):14.

[15] Zhang BG, Deng XH. Oscillation of delay differential equations

on time scales. Math Comput Model 2002;36(11–13):1307–18.

[16] Hardy

GH,

Littlewood

IE,

Polya

G.

Inequalities. Cambridge: University Press; 1959.

Cairo University

Journal of Advanced Research

SHORT COMMUNICATION

On the oscillation of higher order dynamic equations

Said R. Grace

*

Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt

Received 4 March 2012; revised 2 April 2012; accepted 23 April 2012

Available online 2 July 2012

KEYWORDS

Abstract

Oscillation;

Higher Order;

Dynamic equations

aðtÞðx

We present some new criteria for the oscillation of even order dynamic equation

DnÀ1

ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rdcontinuous functions deﬁned on T.

ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

(ii) a, q: T ﬁ R+ = (0, 1) is a real-valued rd-continuous

functions, aD(t) P 0 for t 2 [t0, 1)T and

Introduction

This paper is concerned with the oscillatory behavior of all

solutions of the even order dynamic equation

nÀ1

aðtÞðxD ðtÞÞa

D

þ qðtÞðxðtÞÞa ¼ 0;

ð1:1Þ

on an arbitrary time-scale T ˝ R with Sup T = 1 and n P 2 is

an even integer.

We shall assume that:

(i) a P 1 is the ratio of positive odd integers,

* Tel.: +20 2 35876998.

E-mail address: saidgrace@yahoo.com.

Peer review under responsibility of Cairo University.

Production and hosting by Elsevier

Z

1

aÀ1=a ðsÞDs ¼ 1:

ð1:2Þ

We recall that a solution x of Eq. (1.1) is said to be nonoscillatory if there exists a t0 2 T.

Such that x(t)x(r(t)) > 0 for all t 2 [t0, 1)T; otherwise, it is

said to be oscillatory. Eq. (1.1) is said to be oscillatory if all its

solutions are oscillatory.

The study of dynamic equations on time-scales which goes

back to its founder Hilger [1] as an area of mathematics that

has received a lot of attention. It has been created in order

to unify the study of differential and difference equations.

Recently, there has been an increasing interest in studying

the oscillatory behavior of ﬁrst and second order dynamic

equations on time-scales, see [2–7]. With respect to dynamic

equations on time scales it is fairly new topic and for general

basic ideas and background, we refer to [8,9].

It appears that very little is known regarding the oscillation

of higher order dynamic equations [10–15] and our purpose

here to establish some new criteria for the oscillation criteria

for such equations. The obtained results are new even for the

special cases when T = R and T = Z.

2090-1232 ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.jare.2012.04.003

202

S.R. Grace

Theorem 2.2. Let conditions (i), (ii) and (1.2) hold and

Main results

Z

We shall employ the following well-known lemma.

Z

rðsÞ ðaðsÞÞÀ1

1

t0

Lemma 2.1. [6, Corollary 1]. Assume that n 2 N, s, t 2 T and

f 2 Crd(T, R). Then

Z

t

s

Z

t

ÁÁÁ

gn þ1

¼ ðÀ1Þn

Z

Z

Z

t!1

t

ð2:1Þ

Â

Ã

gðsÞqðsÞ À aðsÞgD ðsÞhÀa

nÀ1 ðs; t0 Þ Ds ¼ 1;

ð2:4Þ

t1

t

hn ðs; rðgÞÞfðgÞDg:

ð2:3Þ

If there exists a positive non-decreasing delta-differentiable

function g such that for every t1 2 [t0, 1)T.

lim sup

gn

1=a

qðuÞDu

Ds ¼ 1:

s

t

fðg1 ÞDg1 Dg2 Á Á Á Dgnþ1

1

then Eq. (1.1) is oscillatory.

s

We shall employ the Kiguradze’s following well-known

lemma.

Theorem 2.1. (Kiguarde’s Lemma [8, Theorem 5]),Let

n 2 N; f 2 Cnrd ðT; RÞ and sup T = 1. Suppose that f is either

n

positive or negative and fD is not identically zero and is either

nonnegative or nonpositive on [t0, 1)T for some t0 2 T. Then,

there

exist

t1 2 [t0, 1)Tm 2 [0, n)Z

such

that

n

ðÀ1ÞnÀm fðtÞfD ðtÞ P 0 holds for all t 2 [t0, 1)T with

j

(I) f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all

j 2 [0, m)z,

j

(II) ðÀ1Þmþj f ðtÞf D ðtÞ > 0 holds for all t 2 [t0, 1)T and all

j 2 [m, n)z

Proof. Let x(t) be a nonoscillatory solution of Eq. (1.1) on

[t0, 1)T. It sufﬁces to discuss the case x is eventually positive

(as – x also solves (1.1) if x does), say x(t) > 0 for t P t1 P t0.

Now, we see that

D

nÀ1

aðtÞðxD ðtÞÞa 6 0 for t P t1 :

nÀ1

It is easy to see that xD ðtÞ > 0 for t P t1 for otherwise,

and by using condition (1.2) we obtain a contradiction to the

fact that x(t) > 0 for t P t1. Now, aD(t) P 0 for t 2 [t0, 1)T.

We have

D

nÀ1

D

nÀ1 a

nÀ1

aðtÞðxD ðtÞÞa ¼ aD ðtÞ xD ðtÞ þ ar ðtÞ ðxD ðtÞÞa

6 0:

The following result will be used to prove the next corollary.

Lemma 2.2. [12, Lemma 2.8].Let supT = 1 and

f 2 Cnrd ðT; RÞðn P 2Þ. Moreover, suppose that Kiguradze’s theon

rem holds with m 2 [1, n)N and fD 6 0 on T. Then there exists a

sufﬁciently large t1 2 T such that

m

fD ðtÞ P hmÀ1 ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :

This implies

nÀ1

ððxD ðtÞÞa ÞD 6 0 for t P t1 :

nÀ1

Next, we let y ¼ xD

rem 1.90] we see that

ð2:2Þ

Z

on [t1, 1)T. From Ref. [9, Theo-

1

ðy þ hlyD ÞaÀ1 dh P ayD

Z

1

The proof of the following corollary follows by an integration

of (2.2).

0 P ðya ÞD ¼ ayD

Corollary 2.1. Assume that the conditions of Lemma 2.2 hold.

Then

Thus we have yD ¼ xD 6 0 on [t1, 1)T and from Theorem 2.1, there exists an integer m 2 {1, 3, . . . , n À 1}. Such that

(I) and (II) hold on [t1, 1)T Clearly xD(t) > 0 for t P t1 and

hence, there exists a constant c > 0 such that

m

fðtÞ P hm ðt; t1 ÞfD ðtÞ for all t 2 ½t1 ; 1ÞT :

Next, we need the following lemma see [16].

Lemma 2.3. If X and Y are nonnegative and k > 1, then

0

yaÀ1 dh

0

¼ ayaÀ1 yD :

n

xðtÞ P c

ð2:5Þ

for t P t1 :

First, we claim that m = n À 1. To this end, we assume that

Xk À kXYkÀ1 þ ðk À 1ÞYk P 0;

xD ðtÞ < 0

where equality holds if and only if X = Y,

Integrating Eq. (1.1) from t P t1 to u P t, letting u ﬁ 1 we

have

It will be convenient to employ the Taylor monomials (see

[9, Section 1.6]) fhn ðt; sÞ1

n¼0 g which are deﬁned recursively by:

Z t

hn ðs; sÞDs; t; s 2 T and n P 1;

hnþ1 ðt; sÞ ¼

nÀ2

nÀ3

and xD ðtÞ > 0

Z

nÀ1

xD ðtÞ P c ðaðtÞÞÀ1

for t P t1 :

1=a

1

for t P t1 :

qðsÞDs

t

s

where it follows that h1(t, s) = t À s but simple formulas in

general do not hold for n P 2.

Now we present the following oscillation results for Eq.

(1.1).

Integrating (2.6) from to and letting ﬁ 1 we get

nÀ2

0 < ÀxD ðtÞ 6 À

Z

t

1

Z

ðaðuÞÞÀ1

u

1=a

1

qðsÞDs

Du:

ð2:6Þ

On the oscillation of higher order dynamic equations

203

Integrating this inequality from t and using condition (2.3)

after Lemma 2.1 we arrive at the desired contradiction. It

follows from Lemma 2.2 with m = n À 1 that

DnÀ1

D

x ðtÞ P hnÀ2 ðt; t1 Þx

ðtÞ for t P t1 ;

ð2:7Þ

and by applying Corollary 2.1 with m = n À 1 instead of Lemma 2.2, we get

nÀ1

xðtÞ P hnÀ1 ðt; t1 ÞxD ðtÞ

D

D

g

g

x

wD 6 Àgq þ r wr À a r

wr :

g

g

x

Using Lemma 2.2 with m = n À 1 in (2.15), we ﬁnd

!

D

nÀ1

g

g

xD

r

wr

hnÀ2

w 6 Àgq þ r w À a r

g

g

x

D

nÀ1

DnÀ1 a

aðx Þ

xa

on ½t1 ; 1ÞT :

ð2:9Þ

g D

g

nÀ1

axD

x

ð2:10Þ

Now set

ð2:11Þ

a

X ¼ ðaðaÀ1=a ÞghnÀ2 Þaþ1

wD ðtÞ 6 ÀgðtÞqðtÞ þ aðtÞgD ðtÞðhnÀ1 ðt; t1 ÞÞÀa ;

¼

t P t1 :

Integrating this inequality from t2 > t1 to t P t2, we have

t

½gðsÞqðsÞ À aðsÞgD ðsÞhÀa

nÀ1 ðs; t1 ÞDs:

ð2:12Þ

t2

Taking upper limit of both sides of the inequality (2.12) as

t ﬁ 1 and using (2.4) we obtain a contradiction to the fact

that w(t) > 0 on [t1, 1)T. This completes the proof. h

Next, we establish the following result.

t!1

Â hnÀ2 ðs; t0 ÞgðsÞ

ð2:13Þ

x þ lhxD

ÃaÀ1

a

ða þ 1Þaþ1

1

ghnÀ2

a

ðgD Þaþ1

on ½t2 ; 1ÞT :

ð2:18Þ

Integrating (2.18) from t2 to t, we get

Z t"

gðsÞqðsÞ À

aðsÞ

1

a

ða þ 1Þaþ1 gðsÞhnÀ2 ðs;t1 Þ

#

ðgD ðsÞÞaþ1 Ds:

Taking upper limit of both sides of (2.19) as t ﬁ 1 and

using (2.13), we obtain a contradiction to the fact that

w(t) > 0 for t P t1 This completes the proof. h

Finally, we present the following interesting result.

Proceeding as in the proof of Theorem 2.1, we obtain

m = n À 1 and (2.7) and (2.8). Deﬁne w as in (2.9) and obtain

(2.10). Now from Ref. [3, Theorem 1.90],

ðxa ÞD ¼ ðaxD Þ

;

ð2:19Þ

Proof. Let x be a nonoscillatory solution of Eq. (1.1), say

x(t) > 0 for t P t1 P t0.

1Â

a !a

!aþ1

and therefore, we ﬁnd

t2

then Eq. (1.1) is oscillatory.

Z

a1=a

aghnÀ2

P 0;

aðsÞðgD ðsÞÞaþ1

aþ1

Ds ¼ 1:

and Y

in Lemma 2.3 with k ¼ aþ1

> 1 to conclude that

a

!

D

g

1

g

a

ðgD Þaþ1

r 1þ1=a

ðw Þ

À r wr þ

a

aþ1

1þ1=a

hnÀ2

g

ga hanÀ2

ða þ 1Þ

ðgr Þ

wðtÞ 6 wðt2 Þ À

ða þ 1Þ

!Àa #

t1

r

w

g

a

a

ðgD Þa

aþ1

wD 6 Àgq þ

Theorem 2.3. Let conditions (i), (ii) (1.2) and (2.3) hold. If

there exists a a positive non-decreasing delta-differentiable

function g such that for every t1 2 [t0, 1)T

1

ð2:17Þ

for t P t2 P t1 ;

Using (2.8) in (2.11), we get

Z t"

lim sup

gðsÞqðsÞ À

on ½t2 ; 1ÞT ;

!

D

g

ðaÞÀ1=a g

r

ðhnÀ2 Þðwr Þ1þ1=a

w 6 Àgq þ r w À a

g

ðgr Þ1þ1=a

ðaðxD Þa Þr þ a ðaðxD Þa ÞD

" x

#

D a

a D

DnÀ1 a r g x À gðx Þ

;

¼ Àgq þ ðaðx Þ Þ

xa ðxr Þa

!a

DnÀ1

D x

6 Àgq þ ag

:

x

Z

1=a

r 1=a

w

w

P

g

g

D

nÀ1

xa

wðtÞ 6 wðt2 Þ À

¼

and thus,

Then on [t1, 1)T, we have

wD ¼

ð2:16Þ

where hnÀ2 = hnÀ2(t, t1). Now we see that

Now, we let

w :¼ g

for t

P t2 P t1 ;

ð2:8Þ

for t P t1 :

ð2:15Þ

dh P axD

0

Z

Theorem 2.4. Let conditions (i), (ii), (1.2) and (2.3) hold. If

there exists a a positive, delta-differentiable function g such that

for every t1 2 [t0, 1)T

Z t"

lim sup

t!1

1

xaÀ1 dh ¼ axaÀ1 xD :

t1

#

1

ðaðsÞÞr ðgD ðsÞÞ2

gðsÞqðsÞ À

Ds

4a gðsÞðhnÀ1 ðs; t1 ÞÞaÀ1 hnÀ2 ðs; t1 Þ

¼ 1;

0

ð2:20Þ

ð2:14Þ

Using (2.14) in (2.10), we have

then Eq. (1.1) is oscillatory.

204

S.R. Grace

Proof. Let x be a nonoscillatoy solution of Eq. (1.1), say

x(t) > 0 for t P t1 P t0

Proceeding as in the proof of Theorem 2.3, we obtain (2.17)

which cam be rewritten as

!

D

g

ðaÞÀ1=a g

r

w 6 Àgq þ r w À a

ðhnÀ2 Þ

g

ðgr Þ1þ1=a

2. We may also employ other types of the time-scales [8,9] e.g.,

T = hZ with h > 0; qN o ; q > 1, T ¼ N 20 , etc. The detail are

left to the reader.

References

D

r 1=aÀ1

Â ðw Þ

r 2

ðw Þ

on ½t2 ; 1ÞT :

ð2:21Þ

Now, using Corollary 2.1 with m = n À 1 we have

x

nÀ1

xD

P hnÀ1 ;

implies on [t2, 1)T that (2.22)

!1Àa

DnÀ1

1=aÀ1

1=aÀ1 1=aÀ1 x

w

¼a

g

x

aÀ1

x

¼ a1=aÀ1 g1=aÀ1 DnÀ1

P a1=aÀ1 g1=aÀ1 haÀ1

nÀ1 :

x

ð2:22Þ

Using (2.22) in (2.21) we have on [t2, 1)T that

wD 6 Àgq þ

D

g

gr

À

¼ Àgq À

þ

aÀ1 r

gðh Þ h

wr À a ðaÞnÀ1r ðgr ÞnÀ2

ðwr Þ2 :

2

r

ahÀ1

nÀ2 ðhnÀ1 Þ

aÀ1

Á1=2

g

À

ððaÞr Þ1=2 gr

r

ðaÞ

ðgD Þ2

4aðhrnÀ1 Þ

6 Àgq þ

aÀ1

hnÀ2

À1Á

4a

!2

ððaÞr Þ1=2 gD

aÀ1

2ðahnÀ2 ðhrnÀ1 Þ

gÞ1=2

:

ðaÞr ðgD Þ2

gðhrnÀ1 Þ

aÀ2

hnÀ2

:

ð2:23Þ

Integrating this inequality from t2 to t, taking upper limit of

the resulting inequality as t ﬁ 1, and applying condition (2.20)

we obtain a contradiction to the fact that w(t) > 0 for t P t1.

This completes the proof.

h

Remarks

1. The results of this paper are presented in a form which is

essentially new and of high degree of generality. Also, we

can easily formulate the above conditions which are new sufﬁcient for the oscillation of Eq. (1.1) on different time-scales

e.g., T = R and T = Z. The details are left to the reader.

[1] Hilger S. Analysis on measure chain-a uniﬁed approach to

contiguous and discrete calculus. Results Math 1990;18:18–56.

[2] Braverman E, Karpuz B. Nonosillation of ﬁrst-order dynamic

equations with several delays. Adv Difference Eq 2010:22. Art.

ID 873459.

[3] Grace SR, Bohner M, Agarwal RP. On the oscillation of second

order half linear dynamic equations. J Differ Eqs Appl

2009;15:451–60.

[4] Grace SR, Agarwal RP, Bohner M, O’Regan D. Oscillation of

second order strongly superlinear and strongly sublinear

dynamic equations. Commun Nonlinear Sci Numer Simulat

2009;14:3463–71.

[5] Sahiner Y. Oscillation of second order delay differential

equations on time-scales. Nonlinear Anal 2005;63(5–7):1073–80.

[6] Bohner M. Some oscillation criteria for ﬁrst order delay

dynamic equations. Far East J Appl Math 2005;18(3):289–304.

[7] Braverman E, Karpuz B. Nonoscillation of second order

dynamic equations with several delays. Abstr Appl Anal

2011:34. Art. ID 591254.

[8] Agarwal RP, Bohner M. Basic calculus on time scales and some

of its applications. Results Math 1999;35(1–2):3–22.

[9] Bohner M, Peterson A. Dynamic Equations on Time-Scales : An

Introduction with Applications. Boston: Birkhauser; 2001.

[10] Chen DX. Oscillation and asymptotic behavior for nth order

nonlinear neutral delay dynamic equations on time scales. Acta

Appl Math 2010;109(3):703–19.

[11] Erbe L, Baoguo J, Peterson A. Oscillation of nth order

superlinear dynamic equations on time scales. Rocky

Mountain J Math 2011;41(2):471–91.

[12] Erbe L, Karpuz B, Peterson A. Kamenev-type oscillation

criteria for higher order neuyral delay dynamic equations. Int

J Differ Eq, in press (2012.-IJDE-1106).

[13] Karpuz B. Asymptotic behavior of bounded solutions of a class

of higher-order neutral dynamic equations. Appl Math Comput

2009;215(6):2174–2183,.

[14] Karpuz B. Unbounded oscillation of higher-order nonlinear

delay dynamic equations of neutral type with oscillating

coefﬁcients. Electon J Qual Theo Differ Eq 2009;9(34):14.

[15] Zhang BG, Deng XH. Oscillation of delay differential equations

on time scales. Math Comput Model 2002;36(11–13):1307–18.

[16] Hardy

GH,

Littlewood

IE,

Polya

G.

Inequalities. Cambridge: University Press; 1959.

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